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2.2. Competition Phenomena 1. 2. 3. 4. 5. Volterra-Lotka Competition Equations Population Dynamics of Fox Rabies in Europe Selection and Evolution of Biological Molecules Laser Beam Competition Equations Rapoport's Model for the Arms Race 2.2.1. Volterra-Lotka Competition Equations Predator-Prey Relationship If N B B N B g B N L N B big (predator) fish N L L N L gL N B N L little (prey) fish g B g L 0 then N B t N B 0 exp B t N L t N L 0 exp L t Solutions to predator-prey problems are usually cyclic with a difference in phase between NB and NL. Rabbits-Foxes Equations r 2 r rf f f rf MF04.nb MF04.mws 02-2.nb Generalizations Limited Resources: Verhulst term NL N L L N L gL N B N L is the saturation number Time-lag: Problem 2-26 2.2.2. Population Dynamics of Fox Rabies in Europe Rabies epidemic in central Europe. Originated in Poland in 1939 Transmitted primarily by the fox population Model 3 categories of fox population: 1. Susceptibles: population density X. Currently health but susceptible to infection. 2. Infected: population density Y Infected, cannot infect the susceptibles. 3. Infectuous: population density Z, Infected, can infect the susceptibles. No recovered category -- high mortality rate X aX b N X XZ Y XZ b N Y N X Y Z Z Y b N Z Symbol Meaning Value a average per capita birth rate 1 yr1 b average per capita natural death rate 0.5 yr1 rabies transmission coefficient 79.67 km2 yr1 inverse of latent period (~28 to 30 days) 13 yr1 death rate rabid foxes (average life expectancy ~ 5 days ) 73 yr1 coefficient of limited food supply 0.1 to 5 km2 yr1 Population Dynamics For Fox Rabies ( X healthy, Y infected foxes ) 0.1 MF05.nb MF05.mws 2.2.3. Selection and Evolution of Biological Molecules Eigen and Schuster : • 1st carriers of genetic information: self-replicating strands of RNA • Mutations: slight errors in the duplication of the nucleotide sequences • Food: energy-rich monomers • Selection- evolution: Darwinian survival-of- the-fittest Symbol Xk(t) Ak Meaning concentration of species k total reproduction rate of species k, including mutations Dk fraction of copies that are precise (quality factor of species k) decomposition (death) rate of species k kl mutation coefficient for producing species k due to errors in the replication of species l Wk Ak Qk Dk net intrinsic rate of producing exact copies Qk Linear rate equation for producing species k X k t Wk X k t N l k 1 kl X l t Since 1 Qk is the fraction of mutations: A 1 Q X k k k k k , l k l kl X l Conservation relation (Eigen and Schuster’s selection criteria) X k n const k d Xk Xk 0 dt k k X k Wk X k Ak 1 Qk X k k k k Ak Qk Dk X k Ak 1 Qk X k k k Ak Dk X k k If Dk > Ak , the species will die out even without competition. Let Dk < Ak , or Ek Ak – Dk > 0 for all k, then X k k Ek X k 0 k “Dilution term” needed to satisfy constraints X k t Wk X k t N l k 1 kl X l t X k 0 X k Assume for simplicity: k 0 Ak Dk X k Ek X k k k E X E t X k k k k k X k t Wk E X k t N l k 1 kl X l t (Quasi-species model) Case N 2 is a Riccati equation: X aX f1 t X f 2 t 0 2 See Problem 2-31 To be solved in Chapter 5 2.2.4. Laser Beam Competition Equations Ruby Laser: Gas cell: 6943A. liquid CCl4 colored with trace I2. Stimulated Thermal Scattering I2 molecules excited by absorbing photons of energy L S De-excitations via collisions with molecules of host liquid. Thermal fluctuations modulation of refractive index of liquid scattering between the laser beams. To establish steady state, duration of light pulses must be long compared to the lifetime of the thermal fluctuations. Laser beams travelling in opposite direction inside cell dI L gI L I S I L dz 0 dI S gI L I S I S dz Laser beams travelling in same direction inside cell dI L gI L I S I L dz dI S gI L I S I S dz Bernoulli equation [see chapter 5] dy f1 z y f 2 y n dz 2.2.5. Rapoport's Model for the Arms Race L.F. Richardson: Defense budgets of European nations for 1909-13. X a1Y Y a2 X aj > 0 For a1 a2 k X t Y t X 0 Y 0 e kt Rapoport Budget growth rates: accelerated in times of crisis decelerated during peace times X m1 X a1Y b1Y 2 Y m2Y a2 X b2 X 2 MF06.nb MF06.mws.